My Favorite Geometry Proof Friday,
August 2, 2013, 1 PM  4:55 PM Organizer: Sarah Mabrouk,
Framingham State University 

This
session invites presenters to share their favorite undergraduate geometry
proofs. These proofs should be suitable for Euclidean and nonEuclidean
geometry courses as well as for courses frequently referred to as “modern” or
“higher” geometry but not those related to differential geometry or
(lowlevel) graduate courses. Proofs must be for theorems other than the
Pythagorean Theorem. Presenters must do the full proof, discuss how the proof
fits into the course, provide information regarding prerequisite topics for
the proof, and discuss associated areas with which students have difficulty
and how such concerns are addressed so that students understand the proof.
Presenters are invited to discuss how they have modified the proof over time
as well as to share historical information for “classic” proofs and
explorations/demonstrations that they use to help students understand the
associated theorem. Abstracts should include the theorem to be
proved/discussed as well as brief background information. 
1 PM  1:15 PM 
Pizzas, Calzones, and
Crusts: Using Symmetry to Slice Up A Circle Michael Nathanson, Saint Mary's College of California My love affair with the socalled Pizza Theorem is now entering
its third decade and continues to provide new joys and unexpected twists,
both for me and my students at both the high school and college levels. The
original theorem shows how to fairly divide the area of a circle in half by
drawing four coincident lines meeting at an arbitrary point on the interior.
I will explain and prove this result using a basic property of perpendicular
chords. I will then show a new dissection proof of how to split a circle
fairly among many people and indicate how this proof provides new insights
into the fair splitting of an annulus, a sphere, and a cantelope.
Whatever that is. The motivation to share this with students is twofold: To
share beautiful mathematical results and ideas and to communicate the process
by which instructive proofs spawn new questions for students to ponder. 
1:20 PM  1:35 PM 
Heron’s Formula: A
Proof Without Words Daniel E. Otero, My current favorite theorem in geometry is a ``proof without
words'' for Heron's formula. This ancient proposition gives the area of a
triangle in the Euclidean plane in terms of the side lengths: a triangle with
side lengths a, b, c has area = [s(sa)(sb)(sc)]^{1/2},
where s= ˝(a+b+c) is its semiperimeter.
Traditional proofs involve a good deal of algebra and/or trigonometry, or the
evaluation of a matrix determinant. The one presented here is based on the
existence of an incircle and a dissection of some simple
figures, so it has a more appropriate geometric “feel” to it. 
1:40 PM  1:55 PM 
Heron’s
Formula for the Area of a Triangle Diana White,
University of Colorado Denver This beautiful result states that the area of a triangle is
the square root of s(sa)(sb)(sc), where s is the semiperimeter (half the sum of the side lengths a, b, and
c). It thus provides a formula for the area of a triangle in terms of just
the side lengths. As such, it has both practical applicability and
mathematical purity. It first appeared in Heron’s Metrica, which was perhaps
written around 75 C.E. Heron’s proof, couched in modern notation and language,
requires only basic knowledge of Euclidean geometry, relying heavily on the
use of similar triangles. Yet the proof is a classic example of “deep”
mathematical arguments that require minimal prerequisite knowledge, reminding
students that sophisticated arguments need not require extensive mathematical
background. Additionally, the proof brings out many key ideas in geometry
that are often not intuitive to those newer to the study of geometry; for
example, inscribing circles in triangles and drawing auxiliary lines. Moreover, in this proof students can readily get “lost”
chasing the symbols in the algebraic steps, seemingly wandering aimlessly
through a series of nonmotivated equations that they are just following
linebyline. By focusing on the drawing, and pointing to it to track the
various steps, students greatly enhances the understanding of the proof. We
discuss how the author guides her students to use this approach to gain a
deeper understanding of both the proof itself as well as when to use diagrams
as a primary focus, and algebraic steps for the formal recordkeeping. 
2 PM  2:15 PM 
Spherical Triangle
Area and Angle Sum Jeff Johannes, SUNY Geneseo In teaching my geometry class, I spend the first month looking
forward to deriving spherical triangle area. Before we get there, students
regularly ask me if they can assume the sum of the angles is a straight
angle, and I just happily anticipate revealing the magic to them. Over the
years, I have enjoyed explaining this argument to several different
audiences, ranging from a general audience with no background to those with a
sophisticated undergraduate training. In this talk we will assume a
familiarity with great circles as straight lines on the sphere to derive a formula
for the area of spherical triangles and infer from it the range of possible
angle sums for triangles on the sphere. 
2:20 PM  2:35 PM 
The Angle Sum Theorem
for Triangles on the Sphere Marshall Whittlesey, California State University San Marcos presentation The angle sum theorem on the sphere states that the sum of the
measures of the angles in a spherical triangle is greater than 180 degrees (p radians). There are a number of ways of proving this theorem.
One standard approach involves the use of tetrahedra
inside the sphere, and another involves showing that the area of a triangle
is proportional to the spherical excess. Here we present a proof which relies
on more basic results of intrinsic spherical geometry rather than on the
extrinsic properties of 3 dimensional space. The
proof resembles a similar standard proof for the angle sum theorem in
hyperbolic geometry. 
2:40 PM  2:55 PM 
The Existence of the
NinePoint Circle for a Given Triangle Stephen Andrilli, Every triangle has an associated “ninepoint circle” which
contains the following nine points: the midpoints of the three sides of the
triangle, the feet of the three altitudes to the triangle, and the midpoints
of the three segments from the vertices of the triangle to the orthocenter of
the triangle. (In special cases, some of these points may coincide.) In this
talk, a proof of this classic result will be given. The "ninepoint
circle" for a given triangle can be constructed in a straightforward
manner using compass and straightedge. A corresponding lab that the author
created using The Geometer's Sketchpad program that constructs the
"ninepoint circle" for a given triangle will be introduced. 
3 PM  3:15 PM 
Ptolemy’s Theorem Pat Touhey, A well known proof of a classic result that is not as well
known as it should be to undergraduate students. We then utilize Ptolemy’s
Theorem and some basic geometry of the circle as a prelude to some
trigonometry. 
3:20 PM – 3:35 PM 
Mini Panel Discussion:
Beneficial Proofs to Include in a Euclidean (Plane) Geometry Course Panelists: Stephen Andrilli, Daniel E. Otero, Pat Touhey, Marshall Whittlesey, Moderator: Sarah Mabrouk,

3:40 PM – 3:55 PM 
Convex Quadrilaterals Braxton Carrigan, Southern CT State University Finding a way to constantly review proof techniques and
concepts previously covered in class for students who have fallen behind but continually
moving forward in Neutral Geometry provides a challenge for any instructor.
This talk will highlight how one theorem can be used to revisit common
misconceptions while still exploring new concepts. Showing that a
quadrilateral ˙ABCD is convex if and
only if $\overline{AC} \cap \overline{BD}$ = Ć can introduce the idea of convex quadrilaterals while
revisiting the relationship between segments, rays, and lines. As a bonus
this theorem can uses symmetry or “without loss of generality” which is often
misused by students. 
4 PM – 4:15 PM 
Quadrature, the
Geometric Mean, Hinged Dissections, and the Purpose of Proof Clark P Wells, Grand
Valley State University Mathematicians will generally agree that proofs are a good
thing (why else would we be talking about our favorites?) and that rigor is
important. But as educators, what is the purpose of proof? I would argue that
in a perfect world a proof should not only verify the truth of a proposition,
but should give insight into the proposition itself. A sad fact is that
proofs often do not give insight, and worse, they can sometimes seem to
students as if they were written to deliberately obscure insight. Sometimes, though,
you can have both rigor and insight. Among my favorite geometry proofs are quadrature proofs, which
I typically discuss in our senior capstone course, The Nature of Modern
Mathematics. The idea of quadrature is to create (typically by compass and straightedge
construction) a square that is “the same size” as a given geometrical object. My very favorite geometry proof is the quadrature of the
rectangle for several reasons. One is that the side length of the square
obtained is the geometric mean of the side lengths of the rectangle, another
is that it can be proven using hinged dissections and then animated using GeoGebra, as I will show in my talk, which leads to
insight about what quadrature and the geometric mean really are. Furthermore,
by taking a theorem due to the ancient Greeks and proving it using modern
technology, I can emphasize the connectedness of mathematical ideas across
centuries. 
4:20 PM – 4:35 PM 
A Simple Proof of the
Classification of Conics by the Discriminant Martin E Flashman, The discriminant of a conic curve in
the real cartesian plane determined by a
nondegenerate quadratic equation of the form Ax^{2} + Bxy +Cy^{2} +Dx +Ey + F = 0 is given by B^{2}  4AC. I will
present a simple proof of the classification of the conic (ellipse, parabola,
or hyperbola) by the discriminant. Many proofs of this result in calculus and
precalculus textbooks are developed using planar rotations. The proof I will
present (without rotations) is based on elementary concepts of homogeneous
coordinates in real projective algebraic geometry. It is suitable for an
upper division geometry course that discusses the interaction of synthetic
and algebraic geometry. 
4:40 PM – 4:55 PM 
It’s Not Hyperbole: A
Transforming Proof Thomas Q Sibley, Nineteenth century geometry transformed our thinking about
geometry, mathematics as a whole and how we see the world  and not just
once, but several times. Hyperbolic, projective and transformational
geometries each reshaped mathematics. In turn a geometry course building on
these topics can open and engage students’ minds. The proof of the theorem
characterizing hyperbolic isometries in the Klein
model provides a capstone for these three areas of geometry. In addition,
this result links this material to spherical geometry and even to Einstein's
theory of relativity. 
This
page was created and is maintained by S. L. Mabrouk, Framingham State
University.
This
page was last modified on Monday, September 02, 2013.